| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 3·9-s − 2·11-s + 5·16-s − 6·18-s + 4·22-s − 4·23-s + 6·25-s + 2·29-s − 6·32-s + 9·36-s − 4·37-s + 8·43-s − 6·44-s + 8·46-s − 12·50-s − 24·53-s − 4·58-s + 7·64-s − 18·67-s + 8·71-s − 12·72-s + 8·74-s − 30·79-s − 16·86-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 9-s − 0.603·11-s + 5/4·16-s − 1.41·18-s + 0.852·22-s − 0.834·23-s + 6/5·25-s + 0.371·29-s − 1.06·32-s + 3/2·36-s − 0.657·37-s + 1.21·43-s − 0.904·44-s + 1.17·46-s − 1.69·50-s − 3.29·53-s − 0.525·58-s + 7/8·64-s − 2.19·67-s + 0.949·71-s − 1.41·72-s + 0.929·74-s − 3.37·79-s − 1.72·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64097690685396419289660444604, −7.56213634158443825645108758370, −7.20513174069266291888622377347, −6.65817966824607617360279018406, −6.15492001133738256684135725184, −5.89723575121056028210205499246, −5.17706309631633075806282122025, −4.51301965109001578483282794481, −4.36076978658625870223006743676, −3.20189338397603369789244092199, −3.15391324087188614918799611445, −2.23521168334471743395036156035, −1.69694527723194007972518751957, −1.05816035148539148989806713716, 0,
1.05816035148539148989806713716, 1.69694527723194007972518751957, 2.23521168334471743395036156035, 3.15391324087188614918799611445, 3.20189338397603369789244092199, 4.36076978658625870223006743676, 4.51301965109001578483282794481, 5.17706309631633075806282122025, 5.89723575121056028210205499246, 6.15492001133738256684135725184, 6.65817966824607617360279018406, 7.20513174069266291888622377347, 7.56213634158443825645108758370, 7.64097690685396419289660444604
